论文标题
平衡II的嵌套球形图。对L层的概括
Nested spheroidal figures of equilibrium II. Generalization to L layers
论文作者
论文摘要
我们提出了一种矢量形式主义，以确定由$ l $同质，刚性旋转的层由球形表面界定的复合体问题的近似解决方案。该方法基于重力参数的重力电位的一阶扩展，从而将论文I中描述的方法推广为$ l = 2 $。对于给定的椭圆相对几何以及在接口处的一组质量密度跳跃，旋转速率和界面压力的序列是通过递归分析获得的。当层以异步旋转的方式旋转时，会产生广泛的平衡，尽管具有负涂抹梯度的构型更有利。相比之下，通过求解$ L-1 $方程的线性系统可以找到的全球旋转状态（所有层都以相同的速率移动）受到了更大的约束。在这种情况下，我们从数学上证明了不允许进行共聚焦和固定构型。得出了小椭圆率的近似公式。这些结果加强并延长已知结果和经典定理，仅限于小椭圆形。与从自洽场方法计算的数值解决方案的比较成功。
We present a vectorial formalism to determine the approximate solutions to the problem of a composite body made of $L$ homogeneous, rigidly rotating layers bounded by spheroidal surfaces. The method is based on the 1st-order expansion of the gravitational potential over confocal parameters, thereby generalizing the method described in Paper I for $L=2$. For a given relative geometry of the ellipses and a given set of mass-density jumps at the interfaces, the sequence of rotation rates and interface pressures is obtained analytically by recursion. A wide range of equilibria result when layers rotate in an asynchronous manner, although configurations with a negative oblateness gradient are more favorable. In contrast, states of global rotation (all layers move at the same rate), found by solving a linear system of $L-1$ equations, are much more constrained. In this case, we mathematically demonstrate that confocal and coelliptical configurations are not permitted. Approximate formula for small ellipticities are derived. These results reinforce and prolongate known results and classical theorems restricted to small elliptiticities. Comparisons with the numerical solutions computed from the Self-Consistent-Field method are successful.
